
to be normalized with a constant \(C\) . The exponent \((N-l)\)  controls the distribution of excited wire magnons. When \(|\beta _{\kappa }|>1\)  (\(|\beta _{\kappa }|<1\) ), the amplitude of \(\psi _{\zeta ,l}\)  decreases (increases) with increasing the sites from 1 to \(N\) , implying the localization at the left (right) edge of the chain.
Figure REF  plots the distribution of the real and imaginary parts of \(\beta _{\pm }\) , which form a loop in the complex plane, under different conditions. When there are net chirality and strong damping of the film, \(|\beta _{\pm }|\)  labeled by the red and blue dots deviate strongly from unit that is indicated by the green dashed line. This is the condition for the emergence of the non-Hermitian skin effect [1]}, [2]}. When the chirality vanishes or the damping of film becomes small, the distribution of \(\beta \)  almost overlaps with the unit circle, indicating the absence of skin effect.
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